Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra and consider the coadjoint action: $$\begin{array}{rcll} &G\times \mathfrak g^\star&\longrightarrow &\mathfrak g^\star\\ &(g,\xi)&\longmapsto & Ad_g^\star(\xi) \end{array}$$ If $\mathcal O^\star$ is a coadjoint orbit, then: $$\begin{array}{rcll} &G\times \mathcal O^\star&\longrightarrow &\mathcal O^\star\\ &(g,\xi)&\longmapsto & Ad_g^\star(\xi) \end{array}$$ must be a transitive action. Therefore the quotient manifold $G/G_\xi$ is diffeomorphic to $\mathcal O^\star$ via: $$g\cdot G_\xi\longmapsto Ad_g^\star(\xi)$$ where $G_\xi$ is the isotropy group of $\xi\in\mathfrak g$.
My question is:
From here, how can we get an isomorphism between $\mathfrak g/\mathfrak g_\xi$ and the tangent space $T_\xi\mathcal O^\star$? ($\mathfrak g_\xi$ is the Lie algebra of $G_\xi$). Maybe taking the induced map?
Many thank!